NON-RIGIDITY IN VAN DER WAALS MOLECULES: SOME CASE STUDIES R G A B O N E " Department of Chemistry, McMaster 1280 Main St West, Hamilton, L8S 4M1, Canada University, Ontario, Weakly bound complexes present significant challenges to both experimentalist and theoretician In many cases, large-amplitude motions of the complexes make the deduction of structural parameters from spectroscopic data uncertain When attempting to calculate those quantities, theoreticians are faced with the consid­ erable cost and difficulty of locating stationary points on shallow potential energy surfaces Often a useful measure of agreement between theory and experiment is the value of a tunneling splitting But accurate calculation of barriers with ab initio methods is far from easy and the use of approximate functional forms to mimic the potential can be similarly fraught with difficulties This chapter describes a number of complexes of acetylene and sulphur dioxide in order to show that nonrigidity may manifest itself in many different ways In each case the consequence is a rich set of spectra whose interpretation caused controversy over a period of years Whilst accompanying theory cannot yet offer quantitative agreement with experiment, its value is in identifying global minima and low-lying transition states from which likely interconversion pathways may be deduced Accompanying group theory can often show that suggested mechanisms are consistent with spectroscopic observations Introduction Van der Waals complexes deserve attention for many reasons Primarily they open a window on the realm of intermolecular forces Additionally, their ex­ istence might direct us to the intermediates and precursors to reactions that form the basis of chemistry in the cosmos and they may play a number of roles in atmospheric phenomena both in comets and small planets The conditions 'out there,' are condusive to the formation of weakly bound species: the low temperatures in regions far from stellar centres and the density of the species of interest make it highly probable that clusters form a measurable compo­ nent of gas composition Additionally, the small molecular species that have been observed in the cosmos are also known to form van der Waals (and/or hydrogen-bonded) complexes in the laboratory The spectroscopic measure­ ment of complexes has typically used micro-wave and infra-red radiation, ' ' a wavelength region which has been successfully used in astronomical mea"Current address: Proteus Molecular Design Ltd., Lyme Green Business Park, Macclesfield, Cheshire, United Kingdom, S K l l OJL 34 Non-Rigidity in van der Waals Molecules 35 surements There is therefore a high expectation that the presence of van der Waals molecules in extraterrestrial settings may be readily confirmed Nevertheless, the elucidaton of their structures has never been straight­ forward, despite the fact that the experimental spectroscopist has a greater level of control than the astronomer over the conditions of observation and the species in question For example, the use of isotopomers has played a critical role in structure-determination by leading to new sets of rotational constants, in the imposition of alternative selection rules on spectral transi­ tions and in gaining insight into intermolecular exchange processes Moreover, in the laboratory, the application of molecular beam methods to the forma­ tion and measurement of van der Waals molecules, whilst enabling a whole menagerie of species to be created and furnishing a wealth of high-precision information on their low-lying states, does not lead to easy comparison with atmospheric or remote measurements Clusters are formed in molecular beams in non-equilibrium conditions and the pressures and temperatures involved are far lower than those typically found in the Solar System Despite such difficulties, evidence for the existence of van der Waals mole­ cules in atmospheres has been gleaned from several sources It is typical to assume a termolecular mechanism of formation and destruction of a dimer, D, from monomers, M, M +M + M r± D + M (1) In which case the third monomer serves to take away excess kinetic energy from the collision Calculations of dimer mole fractions at equilibrium conditions suggest that dimers of nitrogen and oxygen may even be more abundant than 'trace' atmospheric constituents high in the Earth's atmosphere Furthermore, simple models show that the rate of dimer destruction according to Eq is far greater than that due to photo-excitation or vibrational predissociation Indeed, continuum absorption portions of infrared spectra of the Earth's upper atmosphere, conventionally attributed to collision-induced transitions, have in fact been shown to be consistent with transitions amongst internal states of van der Waals dimers Vibrationally averaged effects have always injected uncertainty into the derivation of basic structural parameters, ' and in the case of fluxional be­ haviour, the spectra can be characteristic of the transition state for largeamplitude motion Significant departures from a symmetric configuration can result in enhanced absorption intensity in the vibrational modes being ob­ served The most common result is that spectral lines are split by tunneling between equivalent configurations It will normally require a high-resolution measurement to detect this effect but it manifests itself in almost all modes of 36 R G A Bone vibration Although the barriers to internal motion in van der Waals molecules are extremely small (compared with the binding energies of normal molecules), under the conditions at which they are found, kT is also sufficiently small that the lowest bound states are occupied and large-amplitude motions effectively mix different configurations Of particular difficulty to interpret are the spec­ tra of clusters for whom at least one rotational constant is approximately the same size as a tunneling splitting Traditionally, van der Waals molecules have been described with empirical methods deriving from the long range model of intermolecular forces 9>10>u Methods such as intermolecular perturbation theory 12 are attractive in that they offer an understanding of the forces involved but not easily lead to ac­ curate calculation In the last two decades, as theoretical and computational chemistry have emerged as important fields in all areas of molecular study, it has become common to apply ab initio quantum mechanical methods to van der Waals systems 14,15 Such methods seek to solve the electronic Schrodinger equation at a fixed set of nuclear coordinates, by expanding the wavefunction in a linear combination of atom-centred basis functions 16 ' 17 The least costly way of doing this (without resort to empirical parameterisations of integrals or energy functionals) is to obtain a converged set of molecular orbitals via a selfconsistent field approximation and to add the electronic correlation energy via perturbation theory 18 By now carrying out ab initio calculations of the struc­ tures of van der Waals molecules has successfully masqueraded as a scientific endeavour in its own right and will almost always accompany an experimental investigation There are even examples where structural constants have been reliably computed ab initio without an unambiguous deduction from spec­ tra alone; additionally, calculations may give information on transition state structures and energy barriers which are not always directly derivable from experiment Often, merely an estimate of the overall binding energy can be valuable in thermochemical analysis Once a wavefunction has been computed, a whole host of other useful properties can be computed For example, from the electron density, bonding interactions 19 may be studied using the theory of Atoms in Molecules 20 But ab initio calculations remain difficult for a number of reasons Tech­ nically, it is hard to locate stationary points (minima in transition states) on shallow potential energy surfaces The weakness of the binding forces and the often isotropic nature of the intermolecular pair potentials leads to enor­ mous difficulties in using gradient-based methods to find equilibrium struc­ tures Nevertheless, with advances in computer technology, the CPU require­ ments of such searches have become less demanding Indeed for the species described here, fully optimized geometries were obtainable at adequate levels Non-Rigidity in van der Waals Molecules 37 of theory Despite this, one still has to consider costly methods of calcula­ tion For these molecules, whether dominated by electrostatic forces or not, correlation energy is a necessary component of the calculations, because esti­ mates of the dispersion forces are not possible without it Dispersion forces achieve an overall contraction in optimum intermolecular distances regardless of orientation, even when electrostatic effects are not favourable The need for a correlated level of theory leads to the requirement that large basis sets are used and thus the cost of calculation is driven up A benefit to the use of a large basis set is that the basic electrostatic properties of the individual monomers are likely to be well represented Errors in describing monomer charge distributions can lead to very poor descriptions of complexes A sideeffect of finite basis set methods is the need for some correction for the so-called "Basis Set Superposition Error" (BSSE) 21 This artifact of calculation implies that energy differences between bound complexes and separated monomers are over-estimated The basis set used for the dimer exceeds that used for each separate monomer implying that they not share the same 'zero' of energy in the calculation There is still vociferous debate about the most rigorous way to correct for this problem 22,23 ' 24 ' 25 and basis set quality as well as size influences the effect 26 In the data presented here, BSSE has been taken into account when calculating energy differences but its influence on optimized geometries has not been considered This chapter outlines a number of case-studies of van der Waals molecules which exhibit non-rigid behaviour The examples chosen are from the chem­ istry of SO2 and C2H2 Whilst these are not the most common species either in the interstellar medium or the Earth's atmosphere, they each have a rich variety of cluster formation and exhibit a representative selection of the nonrigid motion that has been observed in the growing catalogue of small van der Waals clusters Consequently they serve as useful prototypes for future inves­ tigations Their study is prefaced by a discussion of interconversion tunneling, symmetry and non-rigidity Tunneling Phenomena and Symmetry When a molecular structure possesses geometric symmetry, 27 there may be more than one equivalent 'version' of it on the potential energy surface 28 In essence, an element of symmetry relates two or more identical nuclei (unless we are considering reflections in the plane containing them) Different versions of a structure arise when distinct labels are attached to atoms of the same element so that more than one unique (non-superimposable) labelling of the structure becomes possible The existence of several versions of a structure (although 38 R G A Bone physically indistinguishable from one another) becomes apparent when mo­ tions between them occur on the timescale of observation 27 Proper symmetry operations of the system now include those permutations which effectively con­ vert one version into another 30 In practice, the lowest barrier on the potential energy surface must not be high enough to quench the effect; a temperature de­ pendence can therefore be expected Examples of equilibrium structures which visit several versions of themselves are widespread in chemistry For example, the NMR spectrum of PF at room temperature shows that all five fluorine atoms are equivalent whereas at low temperatures a pair of 'apical' fluorines distinguishable from the three 'equatorial' fluorines27 of a trigonal bipyramidal minimum energy structure With sufficient energy, a pseudorotation mecha­ nism allows fluorine atoms in each of the two geometrically distinct positions to interchange with one another Similarly, carbocations can be 'scrambled' Bullvalene easily undergoes an internal rearrangement which permutes the car­ bon atoms of its skeleton 31 In inorganic and organic chemistry, such molecules are referred to as 'floppy' or 'fluxional', terms which should be considered syn­ onymous with 'non-rigid' in physical and theoretical chemistry 29 Indeed, in the realm of molecular spectroscopy a 'non-rigid' molecule is one for which interconversion tunneling can be measured 30 Whilst it is usually straightforward to think of a fluxional process in mech­ anistic terms, the quantum mechanical picture is of tunneling between versions Wavefunctions for the different versions penetrate the barriers between them, mix and cause splittings of the levels which are localised in each well Tran­ sitions between such sets of split levels now themselves appear to be split or perturbed So-called interconversion tunneling has been observed in the vibrational and rotational spectra of many molecules and, amongst high-resolution spectroscopists, is a primary indicator of non-rigidity (It should be understood that not all spectroscopic methods will yield direct indication of tunneling in the form of a quantifiable splitting.) The textbook case of tunneling phenomena is the 'umbrella' inversion of ammonia, NH 33 When the three protons have been labelled, there are just two versions of the pyramidal (C^) minimum energy structure: they differ only in the cyclic order of the three hydrogens when viewed in the same sense down the principal axis The potential curve for the motion is a classic doubleminimum The transition state for interconversion is the planar {D3h) struc­ ture (for which there is just one version) In cases where tunneling is observed, the classification and assignment of the spectra has been aided by the Molecular Symmetry (MS) group formal­ ism, introduced by Longuet-Higgins 30 and brought to the spectroscopist by Hougen 34 and Bunker 29 For molecules undergoing interconversion tunnel- Non-Rigidity in van der Waals Molecules 39 ing, the perturbed vibrational levels have symmetry labels derived from the MS group, not the more familiar molecular point group Point group symme­ try operations are either 'proper' (pure rotations) or 'improper' (comprising inversion, reflection or rotation-reflections) 42 It can be shown that proper symmetry operations may be represented by permutations of the labels (and hence coordinates) of identical nuclei whereas improper operations require the product of a space-fixed inversion operation with the appropriate permuta­ tions The MS group expands the set of operations which represent the point group symmetries, to include those permutations and permutation-inversions which represent the interconversion of those versions which are accessible to one another through tunneling 34 Recent work in this area has demonstrated the close relationship between transition state symmetries and the MS group for the tunneling 28 ' 36 In short, the collection of symmetry operations is per­ tinent to any set of accessible structures on the potential energy surface of the motion For complexes of small molecules, we frequently find that there are several possible versions of their equilibrium structures Since it is invariably true that barriers on their intermolecular potential energy surfaces are low, tun­ neling splittings can be a signature of their spectra In many cases, one can hardly dare to assume that a single equilibrium structure has any meaning under the conditions of observation There is by now a long precedent of ob­ serving interconversion tunneling in hydrogen-bonded molecules in terrestrial laboratories The HF dimer is the archetypal example 37 and demonstrates facile exchange of positions of the protons without bond-breaking Similarly, both the water dimer, ' and ammonia dimer, , exhibit complicated in­ terconversion tunneling mechanisms Hydrogen exchange is of acute interest to those studying organic and aqueous systems but the role of tunneling in the positional exchange of heavier elements, whilst often overlooked, is impor­ tant to the study of other chemistries By now, it has become accepted that interconversion tunneling is not exclusive to complexes containing the lighter elements 3.1 Occurrence of Argon, C H and S in Atmospheres and t h e Cosmos Argon Argon is minor component of the Earth's atmosphere, 42 as well as being the third and fourth most abundant constituent of the atmospheres of Mars and Venus, 43 respectively (where C is the dominant species) There are sugges- 40 R G A Bone tions that it is also a measurable component of Titan's atmosphere 3.2 C2H2 Acetylene has not been widely identified in planetary atmospheres, although the presence of small organic moieties in inter-stellar space (both in giant clouds and the grains which go to make up comets) has suggested that it plays a dis­ cernible role The majority of astrophysical searches for C2H2 have focused on the outer planets of the Solar System and their respective moons (particu­ larly Titan) which all have dense organic hazes surrounding them Colussi et al., suggest that dimers of acetylene may be almost non-existent in the upper atmosphere of Titan because of the local conditions: the partial pressure of acetylene is close to its sublimation pressure and the temperature is close to its sublimation point 45 (Additionally, entropic factors mitigate against the formation of larger clusters — trimers, tetramers, etc.) In fact, what clus­ ters of acetylene there are, are likely to be present as ultrafine particles 45 or aerosols ' (This may be a general result: the typical models of dimer for­ mation in equilibrium conditions assume that the temperature is well above that of sublimation 4S Other non-polar species, e.g., CO2 form such parti­ cles 49 ) On the other hand, the presence of methylacetylene and diacetylene in Titan's stratosphere has also been confirmed 50 ' 51 and the appreciation that species trapped in solid particles may undergo photochemical reactions justi­ fies the study of C2H2 complexes Indirect evidence of this is the abundance of methane (a likely photodecomposition product of C2H2) in both Titan's atmosphere and that of Neptune 47 3.3 S02 Present in trace quantities in the Earth's atmosphere 42 (at ppb levels on average), SO2 is certain to participate in acid rain formation Its complexes may make a contribution to chemistry in the higher reaches of the atmosphere either propelled there from volcanic emissions or by upward diffusion from man's industrial processes S0 is a reactive moiety and can be photolytically activated, undergo radical reactions and carry out chemistry in water droplets, as well as form aerosols by reaction with organic species Though a small component of the interstellar medium, SO2 is most likely to be found in the more oxidising planetary atmospheres, in particular the moons of Jupiter and Saturn Ozone, rather than S has been identified on Rhea and Dione 52 whereas S itself has been observed on Europa 53,54 and Callisto 55 ' 56 The first interpretation of the spectra of Europa 53 was that of sulphur ions from the Jovian magnetosphere implanted in water-ice Non-Rigidity in wan der Waals Molecules 41 Ar fl .••* iiViull O^ •0 Figure 1: Ar -so : C, global minimum Nevertheless, subsequent study 54 showed that the spectral data pointed to SO2 ice itself Similarly, spectra on Callisto can be well fit by models of SO2 ice absorption, 5 , allowing for the possibility of mixed water-S02 solids 55 Photo-oxidation of SO2 in low-temperature matrices showed that (S02)2 was a crucial species in SO3 formation Venus's highly oxidising atmosphere has a high proportion of H2SO4 and can therefore be expected to contain a rich sulphur chemistry 4.1 E x a m p l e s of Non-Rigidity in van der Waals Molecules Ar-S02 The Ar—SO2 complex exhibits the simplest case of tunneling: a symmetric double-well potential Spectroscopic studies (both in the microwave 58 ' 59 and the infrared 60 ) affirmed its structure to be a near-prolate asymmetric top with C, symmetry, in which the argon atom sits above the plane of the SO2 molecule (Fig 1) Early ambiguities in deducing the angle, 8, between the vector joining the argon to the SO2 center-of-mass and the SO2 principal axis 58 were resolved in later work, ' finding a value of around 60° But, with fittings to a rigid-rotor model being imperfect, 58 the spectrum was shown to be consistent with an interconversion tunneling mechanism It was suggested that the SO2 monomer rotated about its a-inertial axis, (i.e., maintaining the plane of symmetry which relates the two oxygens to one another throughout its motion), because of the low associated reduced mass 58 Because the overall center-of-mass of the complex stays fixed during the internal rotation, the argon atom effectively travels around the SO2 molecule, sampling two equivalent local minima The tunneling coordinate is therefore 42 R G A Bone Figure 2: Interconversion tunneling potential of Ar—SO3 the angle 8, in Figure The ground state tunneling splitting is 980 MHz which, in a heavy-atom system, was considered to be unusually high, implying that the barrier to inversion is extremely small 58 Indeed, using a WJKB approximation and assuming that the argon atom passes through the ' V of the SO2 rather than around its apex, DeLeon et al 5S deduced a barrier height of just 10 c m - More recent theoretical work, 60 fitting the observed tunneling splittings with a two-dimensional hindering potential (at fixed intermolecular separation) suggested that the barrier is 80 c m - and that an alternative barrier, to orthogonal motion, i.e., with the argon atom in the plane of the SO2 molecule, but alongside the S—0 bond, is above 150 c m - A second calculation, 60 in which electrostatic and induction forces were modelled by atom-sited multipoles and dispersion forces with atom-atom Lennard-Jones coefficients, gave poor quantitative agreement finding that the minimum energy structure was in fact at the Czv structure, the geometry of the experimental minimum being just c m - higher Nevertheless, this empirical model confirmed the supposedly high barrier to internal rotation of the SO2 monomer about its C2axis Consequently there remained considerable uncertainty over the barrier height and the extent of the motion of the argon atom both from experiment and from low-level calculation Extensive ab initio calculations 61 using a correlated method, second order M0ller-Plesset perturbation (MP2) theory, l s established that even a simple Non-Rigidity in van der Waals Molecules 43 four-atom system such as Ar—SO2 has a complicated potential energy surface The global minimum was found to be very similar to the structure deduced from experiment with = 68.5° In addition, four further geometrically dis­ tinct stationary points were identified and motions between them suggested The tunneling coordinate was computed, pointwise using a basis set in which monomer properties are well represented At each value of 0, the angle was kept constant and all other geometric parameters were minimized to ensure that the energy obtained would be truly that of the actual motion The rela­ tive simplicity of the motion makes this an ideal test case for the accuracy of ab initio calculations The potential is represented in Figure One unusual observation was a point of bifurcation at 110° on the tunneling coordinate (i.e., between the global minimum and a planar C2v structure in which the argon atom 'bonds' to the sulphur atom 61 ) The implication is that the argon atom is unlikely to circulate fully around the SO2 molecule along the tunneling coordinate It will either remain trapped in the double well inside the ' V of the SO2 or will skirt off around the 'side' of the SO2 molecule if it has sufficient energy The energy barrier between the C2v transition state and the global min­ imum was calculated to be 59 c m - (17 c m - when corrected for BSSE and zero-point energy effects 61 ) This is in fair agreement with experiment 4-2 (S02)2 The dimer of SO2 proved to be a very hard case for spectroscopists For several years after the first recorded measurement, 62 there was silence in the literature concerning the structure of this species Prediction of the gas-phase structure was made difficult by the realisation that the two monomers were symmetri­ cally inequivalent, i.e., that the orientation of the monomers did not correspond to that found in the crystal structure 63 (Had such an alignment been adopted, the resulting complex would have been nonpolar.) It was deduced from the early microwave measurement 62 that the two monomers changed positions via a not-well understood interconversion tunneling mechanism whose frequency was 70 kHz Only when Taleb-Bendiab et al 64 obtained spectra of many isotopically substituted species did a clearly-defined structure emerge Their model is of a complex with the C, point group, in which the plane of symmetry is the ac-plane with one monomer lying in it (Fig 3) One oxygen atom of the in-plane monomer species acts as an (electron) donor towards the sulphur atom of the other The angle of inclination of the monomer principal axes is 66° Systematic absences in the spectra precluded a accurate estimate of the interconversion tunneling barrier and transition states for the motion could Non-Rigidity in won der Wools Molecules 45 was open to question 65 Even so, a picture emerged in which the Cs structure seemed to be the likely global minimum, (though with the monomer axes in­ clined at an angle differing considerably from the experimental interpretation), and the C, structure would be the lowest lying transition state Visualisation of normal mode motions at both of these structures led to the proposition of a cyclic periodic potential utilising four versions of each structure, (Pig 4) The d symmetry structures allow for the successive inter­ change of positions of 'donor' and 'acceptor' monomers It should be noted, though, that there are possible versions of each of the C, and C, forms This potential is closed and only allows for communication between in each case The potential energy surface of (SC>2)2 is therefore segmented in such a way that groups of versions are separated from one another by a different and probably higher barrier A given version may only interconvert to others using the Cj symmetry transition states It is possible to model this cyclic potential with a simple empirical sinu­ soidal function (following an example set by Fraser et al.68): y=i£V4*(l-cos4fca) (2) k where a is an internal angle such that successive versions of the Cs minimum are at a = 0, TT/2, IT and 37r/2 We add the rotational kinetic energy Hamiltonian, ,2 h d „ (3) = - ^ where / is the b moment of inertia of SO2 and expand solutions in a free rotor basis: T '"» = ^r""°